Thought Provoker
Posts: 530
Joined: April 2007
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Please excuse me for taking so long, but my daughter’s mentor got busy so I quit waiting for him and then I got distracted. But I promised you guys a train wreck so…
Probably the most significant lesson I learned was what the phrase “Jumped the Shark” was all about. I knew about the Happy Days episode, I probably even watched it. I hadn’t made the connection.
Now on to the other stuff.
We started with Special Relativity, Minkowskian geometry and the Twins Paradox.
Here is what Penrose had to say on page 420 of The Road to Reality.
| Quote | | In passing from [Euclidean geometry] to [Minkowskian geometry], there are also changes that relate to inequalities. The most dramatic of these contains the essence of the so-called 'clock paradox' (or 'twin paradox') of special relativity. ... if we accept that the passage of time, as registered by a moving clock, is really a kind of 'arc length' measured along a world line, then the phenomenon is not more puzzling than the path along which this distance is measured. Both are measured by the same formula, namely [integral of ds], but in the Euclidian case, the straight path represents the minimizing of the measured distance between two fixed end-points, whereas in the Minkowski case, it turns out that the straight, i.e. inertial, path represents the maximizing of the measured time between two fixed end events (see also 17.9)." [emphasis Penrose's] |
The path (or arc length) is calculated from the relation…
ds^2 = dt^2 - dx^2 - dy^2 - dz^2
Which means the traveling twin’s path is shorter.
olegt said it simply as…
| Quote | | …a zigzag path in a spacetime with a metric signature (+,-,-,-) has a shorter length than a straight line. Duh! |
However this assumes flat spacetime, which is a “special” case. To be more complete, we need to take into account gravity and curved space. Once we start talking about curved spacetime, references to Minkowskian Geometry confuses the issue. However, we can show that curved space time can be seen as modified flat Minkowskian Geometry.
If we designate “dz” as the up direction in a Earth orbit the spacetime curve will effect the path length calculation as…
ds^2 = Gtt * dt^2 - dx^2 - dy^2 - dz^2 / Gtt
…where the steeper the curve the smaller Gtt. Gtt = 1 for flat space.
Schwarzschild determined that…
Gtt = 1 - Rs / r
Where Rs, the Schwarzschild Radius, is 8.87 millimeters for Earth.
A GPS Satellite in geosynchronous orbit is traveling at 0.0000129217 times the speed of light.
So in one second, the satellite travels 0.0000129217 light-seconds in distance in the X direction.
Calculating the path length for flat space results in…
ds = SQRT(1^2 - 0.0000129217^2 – 0^2 – 0^2)
= 0.999999999916515 seconds
Calculating the path length for r = 2.656175×10^7 of (geosynchronous orbit)
and Gtt = (1 – 0.00887 / 2.656175×10^7) = 0.99999999966606115937391173397837
ds = SQRT(Gtt * 1^2 - 0.0000129217^2 – 0^2 – 0^2 / Gtt)
= 0.999999999749545 seconds of GPS Clock time (slower)
A clock on Earth’s equator is traveling at 0.0000017737252 times the speed of light
So in one second, the clock travels 0.0000017737252 light-seconds in distance in the X direction.
Calculating the path length for r = 6.3781363×10^6 of (Earth’s radius)
and Gtt = (1 – 0.00887 / 6.3781363×10^6) = 0.99999999860931162603094574821175
ds = SQRT(Gtt * 1^2 - 0.0000017737252^2 – 0^2 – 0^2 / Gtt)
= 0.999999999303083 seconds of Earth Clock time (even slower)
Comparing this Earth Clock to the GPS clock results in
Ratio = 0.999999999749545 / 0.999999999303083 = 1.000000000447
Which matches GPS timing observations.
The point being that four dimensional spacetime is real and detectable whether it is a modified Minkowskian Geometry or Schwarzschild Geometry expressed in Cartesian coordinates.
The spacetime path length is zero at the null geodesics (edge of light cone).
I understand this is fairly uncontroversial as long as we are careful with the terminology.
It gets more controversial when General Relativity is merged with Quantum Mechanics.
If 4D spacetime geometry is real, than it is just as real for quantum particles and wavefunctions as it is for traveling twins.
Quantum experiments have demonstrated non-local behavior. Wheeler’s delayed choice and quantum erasure experiments have become so common, they can be done at home inexpensively (thanks Wesley).
One way to describe quantum non-local behavior is to discuss some kind of quantum information communication. But I have learned that the term “quantum information” is too misleading and controversial. Quantum entanglement might be better. Sir Roger Penrose coined the term quanglement.
From Penrose’s book The Road to Reality…
| Quote | | As far as I can make out, quanglement links are always constrained by the light cones, just as are ordinary links, but quanglement links have the novel feature that they can zig-zag backwards and forwards in time, so as to achieve ‘spacelike propagation’. Since quanglement is not information, this does not allow actual signals to be sent faster than light. |
page 603
Now where was I going with this?
Watch out for the curve in the track!
Screech, rumble, crunch, crackle. BOOOOOOM!
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Thank you for all the help and entertainment. I am sorry I didn’t focus as much on this summary as I had intended.
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